# Differential operators on polar harmonic Maass forms and elliptic duality

## TL;DR

This paper investigates polar harmonic Maass forms of negative weight, constructing Poincaré series that span their space and revealing elliptic coefficient duality properties akin to those in harmonic Maass forms.

## Contribution

It introduces a new construction of polar harmonic Maass forms via Poincaré series and uncovers duality properties of their elliptic coefficients.

## Key findings

- Elliptic coefficients of these forms exhibit duality similar to Fourier coefficients.
- Constructed Poincaré series span the space of polar harmonic Maass forms.
- Identified duality properties extend known relationships in harmonic Maass forms.

## Abstract

In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincar\'e series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.